of the cumulative distribution functions and of A and B. If A and B have equal means, B is a mean-preserving spread of A if and only if the area under from minus infinity to is less than or equal to that under from minus infinity to for all real numbers , with strict inequality at some . Both of these mathematical definitions replicate those of second-order stochastic dominance for the case of equal means. Relation to expected utility theory If B is a mean-preserving spread of A then A will be preferred by all expected utility maximizers having concave utility. The converse also holds: if A and B have equal means and A is preferred by all expected utility maximizers having concave utility, then B is a mean-preserving spread of A. See also Stochastic ordering Risk (statistics) Scale parameter References Further reading Theory of probability distributions Decision theory